Some interestin' Cartridges

Here are a few old cartridges from my collection. The .45 S&W and the .45 Colt Government cartridges are identical, except for headstamps, both have the 230gr. Lead bullet. I think these are loaded with LesSmoke powder. The next round is the short case .45 Colt round, while the others are old .45 Colt rounds. Note the lack of cannelure.

Here are two .45 Colt cases sectioned, the balloon head case is one of the nickeled cases, some of the first .45s I ever bought around 1954.

The forerunner of the Boxer primer was already in use by 1873 in commercial ammunition, but the military were concerned about its water-tightness. So alternate priming methods were given a great deal of attention.

The Benet primer was inside primed. The case was drawn copper, closed at the end, sort of like a laboratory test tube. This was then formed to provide a folded rim. The priming compound and anvil were inserted through the mouth of the case, held in place by a cup shaped retainer. This was crimped into place, hence the grooves on the round just ahead of the rim. There was a similar priming method by Martain using a bar instead of the cup. These indentified by having the crimp closer to the rim than on Benet primed cartridges. The problem with inside primed cases was that often the anvil was blown out into the bore when fired, resulting in a bore obstruction.

...[I]s there a museum specifically related to firearms and their history?

I believe that there are several:
There's the NRA museum, of course, in suburban-DC Virginia.
And the Smithsonian has a complete collection of all US military small arms, although it may not always be on display.
And there's the Buffalo Bill Historical Center, in Cody, Wyoming, too.
There's a Colt's collection in Hartford, CT, and the Springfield Armory in Massachusetts.
The Los Angeles County Museum has a nice historic-firearms collection.
And the Metropolitan Museum of Art, in New York, has a superb arms-and-armor collection, specializing in highly-decorated, historic pieces.
There's lots more, but I don't know them all.